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Geometric Constructions
© Copyright 1999, Jim Loy
At the left we see compasses and straightedge. The
compasses are used for drawing arcs and for duplicating lengths. The straight
edge is used (with a pencil or pen) to draw straight lines. There are no marks
on this straight edge (or we ignore the marks that are there). So we do not use
the straight edge to measure lengths. These are the only tools used to do
geometric constructions. And Euclid provided many proofs concerning how to do
things with these tools. Since then, mathematicians have also shown that
certain things cannot be done with these tools. See Trisection Of An Angle.
To the right are a few of the things that can be rather
easily done with these tools. I have constructed an equilateral triangle on a
given base. I have bisected a line segment (a similar construction is to draw a
perpendicular line at a given point, either on the segment or not on the
segment). And I have bisected an angle. We can also duplicate an angle, divide
a line segment into any number of equal-length segments, and quite a few other
things.
Normally, we copy a length by moving the compasses to make two points on
an already drawn line (See Collapsible
Compasses). Similarly, we copy an angle by copying an arc with that central
angle. To draw parallel lines, we duplicate an angle that both lines make with
a transversal (line that intersects both parallels), or in other ways. We can
construct squares, rectangles, parallelograms, and triangles meeting various
conditions.
The diagram on the left shows how to
trisect a line segment (AB). Through A, draw an arbitrary line. On this line
mark off an arbitrary segment, starting at A. Duplicate this segment twice,
producing a trisected segment AC (see the diagram). Draw the line BC. Through
the other two points on the trisected segment, draw lines parallel to BC. These
two lines trisect the segment AB. You can use the same method to divide a
segment into any number of congruent smaller segments.
Addendum #1:
It is easy to construct a
triangle given the lengths of the three sides (diagram on the right), or two
sides and the included angle, or two angles and the included side. Given the
ease of these constructions, it seems strange that the corresponding congruence
"theorems" (Congruence Of Triangles, Part
I and Congruence Of Triangles, Part
II) cannot be proven, except by slightly shady means (superposition).
See The Centers of a Triangle for
clues on how to inscribe a circle in a triangle (bisect two of the angles), and
circumscribe a circle about a triangle (perpendicular bisect two of the sides).
The proofs of these methods are surprisingly simple.
Addendum #2:
One of the things that we cannot legally do is to draw a line to a
circle (or to a line, for that matter). We can only draw lines through two
points, so if you want to draw a line to a circle, you must first find the
proper point on the circle, and then draw the line. The attempted construction
of parallel lines on the left is therefore illegal; we need to draw some
perpendicular lines to find two points, and then draw the line parallel to the
other line. This is just one of the unstated, but reasonable rules which limit
our constructions.
Euclid demonstrated other constructions:
Construct the circle that goes through three given points (diagram on the left,
draw two line segments using these points and the center is where the
perpendicular bisectors of these segments intersect). Through a given point
outside a circle, draw a tangent to the circle (one way is to draw a circle
whose diameter is the segment from the given circle's center and the given
point). A similar construction is to draw a line tangent to two
circles.
Also see these pages concerning constructions:
On the left is a construction of the
Golden Rectangle and the Golden Ratio. On
the right is one way to construct the square root (or square) of a length (The Pythagorean Theorem can also be used to
construct the square root, or square of any given length). Other proportions
and products can be constructed in similar ways, and in other ways (with
similar triangles, for example).
On the left is the construction of two
of the four tangents to two non-intersecting circles. The tangents are parallel
to other more easily constructed lines. Two intersecting circles have two
common tangents, and the construction is similar. And if one circle is inside
the other, there are no common tangents.
It is easy to double a square (construct a square with
double the area). Just use the diagonal as the side of the bigger square. In
the diagram, I have doubled the blue square, producing the yellow square.
Given three points, construct the three
circles with centers at the three points, and externally tangent to the other
circles. In the diagram I have drawn a triangle. The radius of each circle is
the semiperimeter (half the sum of the lengths of the sides of the triangle)
minus the length of the opposite side. That fact (or something similar) makes
the construction easy.
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